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POLYNOMIAL CHAOS EXPANSIONS

The beginnings of Polynomial Chaos (PC) can be traced back to Weiner. He proposed that a spectral expansion of Hermite polynomials in terms of Gaussian random variables can be used to represent certain stochastic processes. Cameron and Martin then demonstrated that the Homogeneous Chaos expansion can be used to approximate any functionals in and that the subsequent approximation converges in . Here is the space of real functions which are continuous on the interval and vanish at . Consequently any second order stochastic process , viewed as a function of the random event , space and time , can be represented by a spectral expansion based upon a trial basis of random Hermite polynomials:

Here the basis, composed of Hermite polynomials of order , is a function of the multidimensional Gaussian random variable with mean zero and unit which are functions of the random parameter

The Hermite Chaos expansion above has been used effectively to solve stochastic differential equations with Gaussian inputs [Xiu 2002, Xiu 2003a, Xiu2003b]. But according to the theorem of Cameron and Martin, the Hermite Chaos expansion will converge for arbitrary second order random processes. For example, Ghanem [ Ghanem 1991, Ghanem 1999] employed Hermite chaos to model log-normal processes. However use of Gaussian processes results in optimal exponential convergence [Lucor 2001]. In some cases, the performance of Hermite Chaos has been shown to decrease substantially when representing random processes with non-Gaussian inputs [Xiu 2002b].

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